Quternionic Ortho Pol

Quaternionic Orthogonal Polynomials for the HP-41

Overview

1°) Legendre Polynomials
2°) Generalized Laguerre's Polynomials
3°) Laguerre's Functions
4°) Hermite Polynomials
5°) Hermite Functions
6°) Chebyshev Polynomials
7°) Chebyshev Functions
8°) UltraSpherical Polynomials
9°) UltraSpherical Functions
10°) Jacobi Polynomials
11°) Jacobi Functions

-The following routines employ the formulae that are used for real variables, which is a natural extension.
-However, I'm not sure that the orthogonality properties still hold for quaternionic arguments...

1°) Legendre Polynomials

Formulae: n.P_n(q) = (2n-1).q.P_n-1(q) - (n-1).P_n-2(q) , P₀(q) = 1 , P₁(q) = q

Data Registers: • R00 = n > 0 ( Register R00 is to be initialized before executing "LEGQ" )

R01 to R04 = q At the end, R05 to R08 = P_n(q) & R09 to R12 = P_n-1(q)
Flags: /
Subroutine: "Q*Q" ( cf "Quaternions for the HP-41" )

-Lines 09 to 16 may be replaced by 6.012 CLRGX if you have an HP-41CX

01 LBL "LEGQ"
02 STO 01
03 RDN
04 STO 02
05 RDN
06 STO 03
07 X<>Y
08 STO 04
09 CLX
10 STO 06
11 STO 07
12 STO 08
13 STO 09
14 STO 10
15 STO 11
16 STO 12

17 SIGN
18 STO 05
19 RCL 00
20 E3
21 /
22 +
23 STO 17
24 LBL 01
25 XEQ "Q*Q"
26 STO 13
27 RDN
28 STO 14
29 RDN
30 STO 15
31 X<>Y
32 STO 16

33 RCL 17
34 INT
35 ENTER^
36 ST+ X
37 1
38 ST- Z
39 -
40 ST* 13
41 ST* 14
42 ST* 15
43 ST* 16
44 X<>Y
45 ST* 09
46 ST* 10
47 ST* 11
48 ST* 12

49 RCL 13
50 X<> 05
51 X<> 09
52 ST- 05
53 RCL 14
54 X<> 06
55 X<> 10
56 ST- 06
57 RCL 15
58 X<> 07
59 X<> 11
60 ST- 07
61 RCL 16
62 X<> 08
63 X<> 12
64 ST- 08

65 RCL 17
66 INT
67 ST/ 05
68 ST/ 06
69 ST/ 07
70 ST/ 08
71 ISG 17
72 GTO 01
73 RCL 08
74 RCL 07
75 RCL 06
76 RCL 05
77 END

( 124 bytes / SIZE 018 )

STACK	INPUTS	OUTPUTS
T	t	t'
Z	z	z'
Y	y	y'
X	x	x'

where P_n(x+y.i+z.j+t.k) = x'+y'.i+z'.j+t'.k with R00 = n integer ( 0 < n < 1000 )

Example: n = 7 q = 1 + i/2 + j/3 + k/4

7 STO 00

   4   1/X
   3   1/X
   2   1/X
   1   XEQ "LEGQ"   >>>>    36.20819589 = R05                         ---Execution time = 25s---
                                 RDN   -51.58337289 = R06
                                 RDN   -34.38891526 = R07
                                 RDN   -25.79168646 = R08

P₇( 1 + i/2 + j/3 + k/4 ) = 36.20819589 - 51.58337289 i - 34.38891526 j - 25.79168646 k and in registers R09 thru R12:

P₆( 1 + i/2 + j/3 + k/4 ) = -9.248135365 - 26.20689563 i - 17.47126375 j - 13.10344782 k

2°) Generalized Laguerre's Polynomials

Formulae: L₀^(a) (q) = 1 , L₁^(a) (q) = a+1-q , n L_n^(a) (q) = (2.n+a-1-q) L_n-1^(a) (x) - (n+a-1) L_n-2^(a) (q)

Data Registers: • R09 = a • R10 = n > 0 ( Registers R09 & R10 are to be initialized before executing "LANQ" )

At the end, R05 to R08 = L_n^(a)(q) & R11 to R14 = L_n-1^(a)(q)
Flags: /
Subroutine: "Q*Q" ( cf "Quaternions for the HP-41" )

01 LBL "LANQ"
02 CHS
03 STO 01
04 RDN
05 CHS
06 STO 02
07 RDN
08 CHS
09 STO 03
10 X<>Y
11 CHS
12 STO 04
13 RCL 09
14 1
15 +
16 ST+ 01
17 CLX

18 STO 06
19 STO 07
20 STO 08
21 STO 11
22 STO 12
23 STO 13
24 STO 14
25 SIGN
26 STO 05
27 RCL 10
28 E3
29 /
30 +
31 STO 00
32 LBL 01
33 XEQ "Q*Q"
34 X<> 05

35 STO 15
36 RDN
37 X<> 06
38 STO 16
39 RDN
40 X<> 07
41 STO 17
42 X<>Y
43 X<> 08
44 STO 18
45 2
46 ST+ 01
47 SIGN
48 RCL 00
49 INT
50 -
51 RCL 09

52 -
53 ST* 11
54 ST* 12
55 ST* 13
56 ST* 14
57 RCL 15
58 X<> 11
59 ST+ 05
60 RCL 16
61 X<> 12
62 ST+ 06
63 RCL 17
64 X<> 13
65 ST+ 07
66 RCL 18
67 X<> 14
68 ST+ 08

69 RCL 00
70 INT
71 ST/ 05
72 ST/ 06
73 ST/ 07
74 ST/ 08
75 ISG 00
76 GTO 01
77 RCL 08
78 RCL 07
79 RCL 06
80 RCL 05
81 END

( 125 bytes / SIZE 019 )

STACK	INPUTS	OUTPUTS
T	t	t'
Z	z	z'
Y	y	y'
X	x	x'

where L_n^(a)(x+y.i+z.j+t.k) = x'+y'.i+z'.j+t'.k with R09 = a & R10 = n integer ( 0 < n < 1000 )

Example: a = sqrt(2) n = 7 q = 1 + 2 i + 3 j + 4 k

2 SQRT STO 09
7 STO 10

   4   ENTER^
   3   ENTER^
   2   ENTER^
   1   XEQ "LANQ"   >>>> 872.2661289 = R05                         ---Execution time = 25s---
                                 RDN    47.12527427 = R06
                                 RDN    70.68791171 = R07
                                 RDN    94.25054886 = R08

L₇^sqrt(2)( 1 + 2 i + 3 j + 4 k ) = 872.2661289 + 47.12527427 i + 70.68791171 j + 94.25054886 k and

L₆^sqrt(2)( 1 + 2 i + 3 j + 4 k ) = 335.6848018 + 123.7427961 i + 185.6141942 j + 247.4855923 k ( in registers R11 thru R14 )

3°) Laguerre's Functions

Formulae: L_n^(a)(q) = [ Gam(n+a+1) / Gam(n+1) / Gam(a+1) ] M(-n,a+1,q)

where Gam = Gamma function
and M = Kummer's function

Data Registers: • R09 = a • R10 = n ( Registers R09 & R10 are to be initialized before executing "LANQ+" )

At the end, R11 to R14 = L_n^(a)(q)
Flags: /
Subroutines:

"KUMQ" ( cf "Quaternionic Special Functions for the HP-41" )
"1/G+" or "GAM+" ... ( cf "Gamma Function for the HP-41" )

01 LBL "LANQ+"
02 X<> 09
03 STO 15
04 X<> 10
05 CHS
06 X<> 09
07 SIGN
08 ST* X

09 ST+ 10
10 X<> L
11 XEQ "KUMQ"
12 1
13 RCL 09
14 -
15 XEQ "1/G+"
16 STO 00

17 RCL 10
18 XEQ "1/G+"
19 ST* 00
20 RCL 10
21 RCL 09
22 -
23 XEQ "1/G+"
24 RCL 00

25 X<>Y
26 /
27 ST* 11
28 ST* 12
29 ST* 13
30 ST* 14
31 RCL 09
32 CHS

33 STO 10
34 RCL 15
35 STO 09
36 RCL 14
37 RCL 13
38 RCL 12
39 RCL 11
40 END

( 81 bytes / SIZE 016 )

STACK	INPUTS	OUTPUTS
T	t	t'
Z	z	z'
Y	y	y'
X	x	x'

where L_n^(a)(x+y.i+z.j+t.k) = x'+y'.i+z'.j+t'.k with R09 = a & R10 = n a & n # -1 , -2 , -3 , ...............

Example: a = sqrt(3) n = PI q = 1 + 2 i + 3 j + 4 k

3 SQRT STO 09
PI STO 10

   4   ENTER^
   3   ENTER^
   2   ENTER^
   1   XEQ "LANQ+"   >>>> -59.37663941 = R11                         ---Execution time = 69s---
                                   RDN     3.135355694 = R12
                                   RDN     4.703033529 = R13
                                   RDN     6.270711372 = R14

L_pi^sqrt(3)( 1 + 2 i + 3 j + 4 k ) = -59.37663941 + 3.135355694 i + 4.703033529 j + 6.270711372 k

4°) Hermite Polynomials

Formulae: H₀(q) = 1 , H₁(q) = 2 q
H_n(q) = 2.q H_n-1(q) - 2 (n-1) H_n-2(q)

Data Registers: • R00 = n > 0 ( Register R00 is to be initialized before executing "HMTQ" )

At the end, R05 to R08 = H_n(q) & R09 to R12 = H_n-1(q)
Flags: /
Subroutine: "Q*Q" ( cf "Quaternions for the HP-41" )

01 LBL "HMTQ"
02 ST+ X
03 STO 01
04 RDN
05 ST+ X
06 STO 02
07 RDN
08 ST+ X
09 STO 03
10 X<>Y
11 ST+ X
12 STO 04
13 CLX
14 STO 06
15 STO 07

16 STO 08
17 STO 09
18 STO 10
19 STO 11
20 STO 12
21 SIGN
22 STO 05
23 RCL 00
24 E3
25 /
26 +
27 STO 17
28 LBL 01
29 XEQ "Q*Q"
30 STO 13

31 RDN
32 STO 14
33 RDN
34 STO 15
35 X<>Y
36 STO 16
37 1
38 RCL 17
39 INT
40 -
41 ST+ X
42 ST* 09
43 ST* 10
44 ST* 11
45 ST* 12

46 RCL 09
47 RCL 13
48 +
49 X<> 05
50 STO 09
51 RCL 10
52 RCL 14
53 +
54 X<> 06
55 STO 10
56 RCL 11
57 RCL 15
58 +
59 X<> 07
60 STO 11

61 RCL 12
62 RCL 16
63 +
64 X<> 08
65 STO 12
66 ISG 17
67 GTO 01
68 RCL 08
69 RCL 07
70 RCL 06
71 RCL 05
72 END

( 105 bytes / SIZE 018 )

STACK	INPUTS	OUTPUTS
T	t	t'
Z	z	z'
Y	y	y'
X	x	x'

where H_n(x+y.i+z.j+t.k) = x'+y'.i+z'.j+t'.k with R00 = n integer ( 0 < n < 1000 )

Example: n = 7 q = 1 + 2 i + 3 j + 4 k

7 STO 00

   4   ENTER^
   3   ENTER^
   2   ENTER^
   1   XEQ "HMTQ"   >>>>      -23716432 = R05                         ---Execution time = 20s---
                                  RDN         -3653024 = R06
                                  RDN         -5479536 = R07
                                  RDN         -7306048 = R08

P₇( 1 + 2 i + 3 j + 4 k ) = -23716432 - 3653024 i - 5479536 j - 7306048 k and in registers R09 thru R12:

P₆( 1 + 2 i + 3 j + 4 k ) = -1122232 + 682816 i + 1024224 j + 1365632 k

5°) Hermite Functions

Formula: H_n(q) = 2ⁿ sqrt(PI) [ (1/Gam((1-n)/2)) M(-n/2,1/2,q²) - ( 2.q / Gam(-n/2) ) M((1-n)/2,3/2,q²) ]

where Gam = Gamma function
and M = Kummer's function

Data Registers: • R00 = n ( Register R00 is to be initialized before executing "HMTQ+" )

At the end, R01 to R04 = H_n(q) R05 to R23: temp
Flags: /
Subroutines:

     "Q*Q"       ( cf "Quaternions for the HP-41" )
     "KUMQ"   ( cf "Quaternionic Special Functions for the HP-41" )
     "1/G+"       ( cf "Gamma Function for the HP-41" )

01 LBL "HMTQ+"
02 STO 01
03 STO 05
04 STO 16
05 RDN
06 STO 02
07 STO 06
08 STO 17
09 RDN
10 STO 03
11 STO 07
12 STO 18
13 X<>Y
14 STO 04
15 STO 08
16 STO 19
17 XEQ "Q*Q"
18 STO 01
19 RDN
20 STO 02
21 RDN
22 STO 03
23 X<>Y

24 STO 04
25 RCL 00
26 STO 15
27 CHS
28 .5
29 STO 10
30 *
31 STO 09
32 RCL 04
33 RCL 03
34 RCL 02
35 RCL 01
36 XEQ "KUMQ"
37 STO 20
38 RDN
39 STO 21
40 RDN
41 STO 22
42 X<>Y
43 STO 23
44 .5
45 ST+ 09
46 RCL 09

47 XEQ "1/G+"
48 ST* 20
49 ST* 21
50 ST* 22
51 ST* 23
52 RCL 04
53 RCL 03
54 RCL 02
55 RCL 01
56 ISG 10
57 XEQ "KUMQ"
58 STO 05
59 RDN
60 STO 06
61 RDN
62 STO 07
63 X<>Y
64 STO 08
65 RCL 16
66 STO 01
67 RCL 17
68 STO 02
69 RCL 18

70 STO 03
71 RCL 19
72 STO 04
73 RCL 15
74 2
75 /
76 CHS
77 XEQ "1/G+"
78 ST+ X
79 CHS
80 ST* 01
81 ST* 02
82 ST* 03
83 ST* 04
84 XEQ "Q*Q"
85 STO 01
86 RDN
87 STO 02
88 RDN
89 STO 03
90 X<>Y
91 STO 04
92 RCL 20

93 ST+ 01
94 RCL 21
95 ST+ 02
96 RCL 22
97 ST+ 03
98 RCL 23
99 ST+ 04
100 2
101 RCL 15
102 STO 00
103 Y^X
104 PI
105 SQRT
106 *
107 ST* 01
108 ST* 02
109 ST* 03
110 ST* 04
111 RCL 04
112 RCL 03
113 RCL 02
114 RCL 01
115 END

( 190 bytes / SIZE 024)

STACK	INPUTS	OUTPUTS
T	t	t'
Z	z	z'
Y	y	y'
X	x	x'

where H_n(x+y.i+z.j+t.k) = x'+y'.i+z'.j+t'.k with R00 = n

Example: n = PI q = 1 + i/2 + j/3 + k/4

PI STO 00

   4   1/X
   3   1/X
   2   1/X
   1   XEQ "HMTQ+"   >>>> -17.81760491 = R01                         ---Execution time = 100s---
                                    RDN    2.845496140 = R02
                                    RDN    1.896997429 = R03
                                    RDN    1.422748073 = R04

H_pi( 1 + i/2 + j/3 + k/4 ) = -17.81760491 + 2.845496140 i + 1.896997429 j + 1.422748073 k

6°) Chebyshev Polynomials

Formulae:

CF 02 T_n(q) = 2q.T_n-1(q) - T_n-2(q) ; T₀(q) = 1 ; T₁(q) = q ( first kind )
SF 02 U_n(q) = 2q.U_n-1(q) - U_n-2(q) ; U₀(q) = 1 ; U₁(q) = 2q ( second kind )

Data Registers: • R00 = n ( Register R00 is to be initialized before executing "CHBQ" )

When the program stops, T_n(q) or U_n(q) are in registers R05 thru R08
T_n-1(q) or U_n-1(q) are in registers R09 thru R12

Flag: F02 if F02 is clear, "CHBQ" calculates the Chebyshev polynomials of the first kind: T_n(q)
if F02 is set, "CHBQ" calculates the Chebyshev polynomials of the second kind: U_n(q)

Subroutine: "Q*Q" ( cf "Quaternions for the HP-41" )

01 LBL "CHBQ"
02 STO 01
03 ST+ 01
04 FS? 02
05 CLX
06 STO 09
07 RDN
08 STO 02
09 ST+ 02
10 FS? 02
11 CLX
12 STO 10
13 RDN

14 STO 03
15 ST+ 03
16 FS? 02
17 CLX
18 STO 11
19 X<>Y
20 STO 04
21 ST+ 04
22 FS? 02
23 CLX
24 STO 12
25 CLX
26 STO 06

27 STO 07
28 STO 08
29 SIGN
30 STO 05
31 RCL 00
32 E3
33 /
34 +
35 STO 13
36 LBL 01
37 XEQ "Q*Q"
38 ST- 09
39 X<> 09

40 CHS
41 X<> 05
42 STO 09
43 X<> 10
44 -
45 X<> 06
46 STO 10
47 X<> 11
48 -
49 X<> 07
50 STO 11
51 X<> 12
52 -

53 X<> 08
54 STO 12
55 ISG 13
56 GTO 01
57 RCL 08
58 RCL 07
59 RCL 06
60 RCL 05
61 END

( 94 bytes / SIZE 013 )

STACK	INPUTS	OUTPUTS
T	t	t'
Z	z	z'
Y	y	y'
X	x	x'

where T_n(x+y.i+z.j+t.k) = x'+y'.i+z'.j+t'.k if CF 02 with R00 = n integer ( 0 < n < 1000 )
or U_n(x+y.i+z.j+t.k) = x'+y'.i+z'.j+t'.k if SF 02

Example: n = 7 q = 1 + 2 i + 3 j + 4 k

7 STO 00

• CF 02 Chebyshev Polynomials 1st kind

   4   ENTER^
   3   ENTER^
   2   ENTER^
   1   XEQ "CHBQ"   >>>>       -9524759                          ---Execution time = 16s---
                                  RDN       -1117678
                                  RDN       -1676517
                                  RDN       -2235356

T₇( 1 + 2 i + 3 j + 4 k ) = -9524759 - 1117678 i - 1676517 j - 2235356 k

• SF 02 Chebyshev Polynomials 2nd kind

   4   ENTER^
   3   ENTER^
   2   ENTER^
   1   XEQ "CHBQ"   >>>>     -18921448                          ---Execution time = 16s---
                                  RDN       -2198096
                                  RDN       -3297144
                                  RDN       -4396192

U₇( 1 + 2 i + 3 j + 4 k ) = -18921448 - 2198096 i - 3297144 j - 4396192 k

7°) Chebyshev Functions

Formulae:

CF 02 T_n(cos Q) = cos n.Q ( first kind ) with cos Q = q
SF 02 U_n(cos Q) = [ sin (n+1).Q ] / sin Q ( second kind )

Data Registers: • R00 = n ( Register R00 is to be initialized before executing "CHBQ+" )

R01 thru R09: temp

Flag: F02 if F02 is clear, "CHBQ+" calculates the Chebyshev function of the first kind: T_n(q)
if F02 is set, "CHBQ+" calculates the Chebyshev function of the second kind: U_n(q)

Subroutine: "Q*Q" "ACOSQ" "COSQ" "SINQ" "1/Q" ( cf "Quaternions for the HP-41" )

01 LBL "CHBQ+"
02 X<> 00
03 STO 09
04 X<> 00
05 XEQ "ACOSQ"
06 FS? 02
07 GTO 00
08 X<> 09
09 STO 00
10 ST* 09
11 ST* Y
12 ST* Z

13 ST* T
14 X<> 09
15 XEQ "COSQ"
16 RTN
17 LBL 00
18 STO 05
19 RDN
20 STO 06
21 RDN
22 STO 07
23 RDN
24 STO 08

25 RDN
26 XEQ "SINQ"
27 XEQ "1/Q"
28 STO 01
29 RDN
30 STO 02
31 RDN
32 STO 03
33 X<>Y
34 STO 04
35 RCL 09
36 STO 00

37 1
38 +
39 ST* 05
40 ST* 06
41 ST* 07
42 RCL 08
43 *
44 RCL 07
45 RCL 06
46 RCL 05
47 XEQ "SINQ"
48 STO 05

49 RDN
50 STO 06
51 RDN
52 STO 07
53 X<>Y
54 STO 08
55 XEQ "Q*Q"
56 END

( 107 bytes / SIZE 010 )

STACK	INPUTS	OUTPUTS
T	t	t'
Z	z	z'
Y	y	y'
X	x	x'

where T_n(x+y.i+z.j+t.k) = x'+y'.i+z'.j+t'.k if CF 02 with R00 = n
or U_n(x+y.i+z.j+t.k) = x'+y'.i+z'.j+t'.k if SF 02

Example: n = PI q = 0.1 + 0.2 i + 0.3 j + 0.4 k

PI STO 00

• CF 02 Chebyshev Function 1st kind

   0.4   ENTER^
   0.3   ENTER^
   0.2   ENTER^
   0.1   XEQ "CHBQ+"   >>>>     -0.143159414                          ---Execution time = 22s---
                                      RDN       -0.905090620
                                      RDN       -1.357635929
                                      RDN       -1.810181239

T_PI( 0.1 + 0.2 i + 0.3 j + 0.4 k ) = -0.143159414 - 0.905090620 i - 1.357635929 j - 1.810181239 k

• SF 02 Chebyshev Function 2nd kind

   0.4   ENTER^
   0.3   ENTER^
   0.2   ENTER^
   0.1   XEQ "CHBQ+"   >>>>      -0.386199512                          ---Execution time = 29s---
                                      RDN       -1.369695950
                                      RDN       -2.054543924
                                      RDN       -2.739391897

U_PI( 0.1 + 0.2 i + 0.3 j + 0.4 k ) = -0.386199512 - 1.369695950 i - 2.054543924 j - 2.739391897 k

8°) UltraSpherical Polynomials

Formulae: C₀^(a) (q) = 1 ; C₁^(a) (q) = 2.a.q ; (n+1).C_n+1^(a) (q) = 2.(n+a).x.C_n^(a) (q) - (n+2a-1).C_n-1^(a) (q) if a # 0

C_n⁽⁰⁾ (q) = (2/n).T_n(q)

Data Registers: • R09 = a • R10 = n > 0 ( Registers R09 & R10 are to be initialized before executing "USPQ" )

At the end, R05 to R08 = C_n^(a)(q)
& R11 to R14 = C_n-1^(a)(q) if a # 0

Flag: F02 if a = 0
Subroutines: "Q*Q" ( cf "Quaternions for the HP-41" ) & "CHBQ" if a = 0

01 LBL "USPQ"
02 STO 01
03 RDN
04 STO 02
05 RDN
06 STO 03
07 X<>Y
08 STO 04
09 RCL 09
10 X#0?
11 GTO 00
12 RCL 10
13 STO 00
14 RCL 04
15 RCL 03
16 RCL 02
17 RCL 01
18 CF 02
19 XEQ "CHBQ"
20 CLX
21 STO 09
22 2

23 RCL 00
24 STO 10
25 /
26 ST* 05
27 ST* 06
28 ST* 07
29 ST* 08
30 GTO 02
31 LBL 00
32 CLX
33 STO 06
34 STO 07
35 STO 08
36 STO 11
37 STO 12
38 STO 13
39 STO 14
40 SIGN
41 STO 05
42 RCL 10
43 E3
44 /

45 +
46 STO 00
47 LBL 01
48 XEQ "Q*Q"
49 STO 15
50 RDN
51 STO 16
52 RDN
53 STO 17
54 X<>Y
55 STO 18
56 RCL 09
57 1
58 -
59 STO 19
60 RCL 00
61 INT
62 +
63 ST+ X
64 ST* 15
65 ST* 16
66 ST* 17

67 ST* 18
68 RCL 00
69 INT
70 RCL 19
71 ST+ X
72 +
73 ST* 11
74 ST* 12
75 ST* 13
76 ST* 14
77 RCL 15
78 RCL 05
79 X<> 11
80 -
81 STO 05
82 RCL 16
83 RCL 06
84 X<> 12
85 -
86 STO 06
87 RCL 17
88 RCL 07

89 X<> 13
90 -
91 STO 07
92 RCL 18
93 RCL 08
94 X<> 14
95 -
96 STO 08
97 RCL 00
98 INT
99 ST/ 05
100 ST/ 06
101 ST/ 07
102 ST/ 08
103 ISG 00
104 GTO 01
105 LBL 02
106 RCL 08
107 RCL 07
108 RCL 06
109 RCL 05
110 END

( 164 bytes / SIZE 020 )

STACK	INPUTS	OUTPUTS
T	t	t'
Z	z	z'
Y	y	y'
X	x	x'

where C_n^(a)(x+y.i+z.j+t.k) = x'+y'.i+z'.j+t'.k with R09 = a & R10 = n integer ( 0 < n < 1000 )

Example: a = sqrt(2) n = 7 q = 1 + i/2 + j/3 + k/4

2 SQRT STO 09
7 STO 10

   4   1/X
   3   1/X
   2   1/X
   1   XEQ "USPQ"   >>>>    324.5443961                         ---Execution time = 28s---
                                 RDN   -689.5883609
                                 RDN   -459.7255744
                                 RDN   -344.7941813

C₇^sqrt(2)( 1 + i/2 + j/3 + k/4 ) = 324.5443961 - 689.5883609 i - 459.7255744 j - 344.7941813 k

-Likewise, C₇⁽⁰⁾( 1 + i/2 + j/3 + k/4 ) = 29.24554794 - 33.27910357 i - 22.18606906 j - 16.63955179 k ( in 19 seconds )

9°) UltraSpherical Functions

Formulae: Assuming a # 0

C_n^(a)(q) = [ Gam(n+2a) / Gam(n+1) / Gam(2a) ] ₂F₁( -n , n+2a , n+1/2 , (1-q)/2 )

where ₂F₁ is the hypergeometric function and Gam = Gamma function

Data Registers: • R09 = a # 0 • R10 = n ( Registers R09 & R10 are to be initialized before executing "USPQ+" )

At the end, R12 to R15 = C_n^(a)(q)

Flags: /
Subroutines:

"HGFQ" ( cf "Quaternionic Special Functions for the HP-41" )
"1/G+" or "GAM+" ( cf "Gamma Function for the HP-41" )

01 LBL "USPQ+"
02 STO 01
03 RDN
04 STO 02
05 RDN
06 STO 03
07 X<>Y
08 STO 04
09 RCL 09
10 STO 11
11 STO 16

12 ST+ X
13 RCL 10
14 CHS
15 STO 09
16 -
17 STO 10
18 .5
19 ST+ 11
20 CHS
21 ST* 01
22 ST* 02

23 ST* 03
24 ST* 04
25 ST- 01
26 RCL 04
27 RCL 03
28 RCL 02
29 RCL 01
30 XEQ "HGFQ"
31 RCL 10
32 XEQ "1/G+"
33 STO 00

34 RCL 09
35 CHS
36 STO 10
37 1
38 +
39 XEQ "1/G+"
40 ST/ 00
41 RCL 16
42 STO 09
43 ST+ X
44 XEQ "1/G+"

45 RCL 00
46 /
47 ST* 12
48 ST* 13
49 ST* 14
50 ST* 15
51 RCL 15
52 RCL 14
53 RCL 13
54 RCL 12
55 END

( 89 bytes / SIZE 017 )

STACK	INPUTS	OUTPUTS
T	t	t'
Z	z	z'
Y	y	y'
X	x	x'

where C_n^(a)(x+y.i+z.j+t.k) = x'+y'.i+z'.j+t'.k with R09 = a # 0 & R10 = n

Example: a = sqrt(2) n = sqrt(3) q = 1 + i/2 + j/3 + k/4

2 SQRT STO 09
3 SQRT STO 10

   4   1/X
   3   1/X
   2   1/X
   1   XEQ "USPQ"   >>>>    3.241115510                         ---Execution time = 77s---
                                 RDN    4.848277901
                                 RDN    3.232185263
                                 RDN    2.424138947

C_sqrt(3)^sqrt(2)( 1 + i/2 + j/3 + k/4 ) = 3.241115510 + 4.848277901 i + 3.232185263 j + 2.424138947 k

10°) Jacobi Polynomials

Formulae: P₀^(a;b) (q) = 1 ; P₁^(a;b) (q) = (a-b)/2 + q (a+b+2)/2

2n(n+a+b)(2n+a+b-2) P_n^(a;b) (q) = [ (2n+a+b-1).(a²-b²) + q (2n+a+b-2)(2n+a+b-1)(2n+a+b) ] P_n-1^(a;b) (q)
- 2(n+a-1)(n+b-1)(2n+a+b) P_n-2^(a;b) (q)

Data Registers: R00 thru R22: temp ( Registers R09-R10-R11 are to be initialized before executing "PABNQ" )

                                 • R09 = a
                                 • R10 = b
                                 • R11 = n > 0

At the end, R05 to R08 = P_n^(a;b)(q) R16 to R19 = q
& R12 to R15 = P_n-1^(a;b)(q)

Flags: /
Subroutine: "Q*Q" ( cf "Quaternions for the HP-41" )

-Lines 48 & 137 are three-byte GTOs
-Lines 129 to 136 may be replaced by 16.001004 REGMOVE

01 LBL "PABNQ"
02 STO 01
03 STO 05
04 STO 16
05 RDN
06 STO 02
07 STO 06
08 STO 17
09 RDN
10 STO 03
11 STO 07
12 STO 18
13 X<>Y
14 STO 04
15 STO 08
16 STO 19
17 CLX
18 STO 13
19 STO 14
20 STO 15
21 SIGN
22 STO 12
23 RCL 11
24 E3
25 /
26 +
27 STO 00
28 RCL 09
29 RCL 10

30 +
31 STO 22
32 2
33 ST+ Y
34 /
35 ST* 05
36 ST* 06
37 ST* 07
38 ST* 08
39 RCL 09
40 RCL 10
41 -
42 2
43 /
44 ST+ 05
45 LBL 01
46 ISG 00
47 FS? 30
48 GTO 00
49 RCL 22
50 RCL 00
51 INT
52 ST+ X
53 +
54 STO 20
55 RCL 00
56 INT
57 RCL 09
58 +

59 1
60 -
61 STO 21
62 *
63 RCL 00
64 INT
65 RCL 10
66 +
67 1
68 -
69 ST+ 21
70 *
71 ST+ X
72 CHS
73 ST* 12
74 ST* 13
75 ST* 14
76 ST* 15
77 RCL 20
78 STO Y
79 1
80 -
81 STO 20
82 *
83 RCL 21
84 *
85 ST* 01
86 ST* 02
87 ST* 03

88 ST* 04
89 RCL 20
90 RCL 09
91 X^2
92 RCL 10
93 X^2
94 -
95 *
96 ST+ 01
97 XEQ "Q*Q"
98 ST+ 12
99 RDN
100 ST+ 13
101 RDN
102 ST+ 14
103 X<>Y
104 ST+ 15
105 RCL 21
106 RCL 22
107 RCL 00
108 INT
109 ST+ Y
110 ST+ X
111 *
112 *
113 ST/ 12
114 ST/ 13
115 ST/ 14
116 ST/ 15

117 RCL 05
118 X<> 12
119 STO 05
120 RCL 06
121 X<>13
122 STO 06
123 RCL 07
124 X<>14
125 STO 07
126 RCL 08
127 X<> 15
128 STO 08
129 RCL 16
130 STO 01
131 RCL 17
132 STO 02
133 RCL 18
134 STO 03
135 RCL 19
136 STO 04
137 GTO 01
138 LBL 00
139 RCL 08
140 RCL 07
141 RCL 06
142 RCL 05
143 END

( 215 bytes / SIZE 023 )

STACK	INPUTS	OUTPUTS
T	t	t'
Z	z	z'
Y	y	y'
X	x	x'

where P_n^(a;b)(x+y.i+z.j+t.k) = x'+y'.i+z'.j+t'.k with R09 = a , R10 = b , R11 = n ( n integer 0 < n < 1000 )

Example: a = sqrt(2) b = sqrt(3) n = 7 q = 1 + i/2 + j/3 + k/4

   2   SQRT   STO 09
   3   SQRT   STO 10
   7   STO 11

   4   1/X
   3   1/X
   2   1/X
   1   XEQ "PABNQ"   >>>>   143.5304515                         ---Execution time = 34s---
                                    RDN   -310.8681606
                                    RDN   -207.2454402
                                    RDN   -155.4340801

P₇^{sqrt(2);sqrt(3)}( 1 + i/2 + j/3 + k/4 ) = 143.5304515 - 310.8681606 i - 207.2454402 j - 155.4340801 k

11°) Jacobi Functions

~
Formula: P_n^(a;b) (q) = [ Gam(a+n+1) / Gam(n+1) ] ₂F₁ ( -n , a+b+n+1 , a+1 , (1-q)/2 )

where ₂F₁ tilde is the regularized hypergeometric function

Data Registers: R00 thru R17: temp ( Registers R09-R10-R11 are to be initialized before executing "JCPQ" )

                                 • R09 = a
                                 • R10 = b
                                 • R11 = n

At the end, R12 to R15 = P_n^(a;b) (q)

Flag: F00
Subroutines:

"HGFQ" ( cf "Quaternionic Special Functions for the HP-41" )
"1/G+" or "GAM+" ( cf "Gamma Function for the HP-41" )

01 LBL "JCPQ"
02 STO 01
03 RDN
04 STO 02
05 RDN
06 STO 03
07 X<>Y
08 STO 04
09 2
10 1/X
11 CHS
12 ST* 01
13 ST* 02
14 ST* 03
15 ST* 04

16 ST- 01
17 RCL 11
18 ENTER^
19 CHS
20 X<> 09
21 STO 11
22 STO 16
23 RCL 10
24 STO 17
25 +
26 +
27 1
28 ST+ 11
29 +
30 STO 10

31 RCL 04
32 RCL 03
33 RCL 02
34 RCL 01
35 XEQ "HGFQ"
36 FS?C 00
37 GTO 00
38 RCL 11
39 XEQ "1/G+"
40 ST* 12
41 ST* 13
42 ST* 14
43 ST* 15
44 LBL 00
45 1

46 RCL 09
47 -
48 XEQ "1/G+"
49 STO 00
50 RCL 09
51 CHS
52 ENTER^
53 X<> 11
54 +
55 XEQ "1/G+"
56 RCL 00
57 X<>Y
58 /
59 ST* 12
60 ST* 13

61 ST* 14
62 ST* 15
63 RCL 16
64 STO 09
65 RCL 17
66 STO 10
67 RCL 15
68 RCL 14
69 RCL 13
70 RCL 12
71 END

( 122 bytes / SIZE 018 )

STACK	INPUTS	OUTPUTS
T	t	t'
Z	z	z'
Y	y	y'
X	x	x'

where P_n^(a;b)(x+y.i+z.j+t.k) = x'+y'.i+z'.j+t'.k with R09 = a , R10 = b , R11 = n

Examples: a = sqrt(2) b = sqrt(3) n = PI q = 1 + i/2 + j/3 + k/4

2 SQRT STO 09
3 SQRT STO 10
PI STO 11

   4   1/X
   3   1/X
   2   1/X
   1   XEQ "JCPQ"   >>>>   -10.14397526                         ---Execution time = 73s---
                                RDN     11.74139998
                                RDN       7.827599989
                                RDN       5.870699990

P_PI^{sqrt(2);sqrt(3)}( 1 + i/2 + j/3 + k/4 ) = -10.14397526 + 11.74139998 i + 7.827599989 j + 5.870699990 k

-Likewise, P_PI^(-4;+4)( 1 + i/2 + j/3 + k/4 ) = 0.360213558 - 0.125970281 i - 0.083980187 j - 0.062985140 k ( in 109 seconds )

Note:

-Unless the function is a polynomial, the series is convergent if | 1 - q | < 2

hp41programs

Quaternionic Orthogonal Polynomials for the HP-41